Algebras of slowly growing length

TL;DR

This paper explores a broad class of finite-dimensional algebras with slowly growing length, establishing upper bounds and including many classical algebra types like Lie, Leibniz, Novikov, and Zinbiel algebras.

Contribution

It extends the concept of characteristic sequences to non-unital algebras and provides polynomial conditions ensuring slow length growth, with explicit upper bounds.

Findings

Finite-dimensional Lie, Leibniz, Novikov, Zinbiel algebras have length ≤ their dimension.

Derived polynomial conditions for slow length growth.

Established exact upper bounds for the length of these algebras.

Abstract

We investigate the class of finite dimensional not necessary associative algebras that have slowly growing length, that is, for any algebra in this class its length is less than or equal to its dimension. We show that this class is considerably big, in particular, finite dimensional Lie algebras as well as many other important classical finite dimensional algebras belong to this class, for example, Leibniz algebras, Novikov algebras, and Zinbiel algebras. An exact upper bounds for the length of these algebras is proved. To do this we transfer the method of characteristic sequences to non-unital algebras and find certain polynomial conditions on the algebra elements that guarantee the slow growth of the length function.